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Let $A \in F^{11 \times10}$ and $B\in$ $F^{10\times11}$

We only know $2$ rows of $A$ and $3$ columns of $B$. How many entries of $B\cdot A$ can we know?

I think the answer is none, because there are dot product, row-row, column-column operations but no column-row one

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2 Answers 2

up vote 5 down vote accepted

The $(i,j)$th entry of $BA$ is the $i$th row of $B$ times the $j$th column of $A$. Because we don't know any of the rows of $B$ completely (and we don't know any of the columns of $A$ completely), we don't know any of the entries of $BA$.

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Hint: This is a picture of matrix multiplication enter image description here

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But the question is asking about $BA$, not $AB$. –  littleO Jul 22 at 5:23
    
@LittleO This picture shows also that matrix multiplication between $A$ and $B$ makes sense only if the number of element in a row of $A$ are the same as the number of element in the column of $B$, therefore if this property is not satisfied you can't multiply two matrices. :) –  Ale Jul 22 at 5:33
    
Well, your answer is a good hint because it shows how matrix multiplication works. –  littleO Jul 22 at 5:34

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